Fundamentals of Electromagnetism

• Physical system: a physical system in electromagnetism can be as simple as a single electron or as complex as an entire power grid.
  • Point Charge System: A single charged particle (like an electron) or a collection of charges.
  • Current-Carrying Conductors: Wires carrying current, solenoids, or circuits.
  • Electromagnetic Fields: The electric field and magnetic field that exist in space and interact with matter.
  • Electromagnetic Waves: Light, radio waves, or any other form of electromagnetic radiation.
  • Capacitor and Inductor Systems: Systems storing energy in electric and magnetic fields.
  • Plasma: Ionized gases where charged particles move freely, interacting with electric and magnetic fields.
• State of the physical system: defined by the physical quantities that describe the system at a given moment in time (t) and space (x).
  • Electomagnetic fields: $E(x,t), B(x,t)$ strength as a function of space and time
  • Propertie of the physical object/system:
    • electric charges ($q, Q$), current ($dQ/dt$), current density ($\mathbf{J}$)
    • Charge distribution : charges located per unit space
      • charge density ($dQ/dx$)
    • Electromagnetic potential: scalar ($V(x,t)$), vector ($A$)
    • Electromagnetic wave, electromagnetic fields $E,B$
    • mass ($m$), velocity $(v)$, position $(x,y,z)$
    • frequency, wavelength, polarization, amplitude
    • Energy density and Momentum
  • A physical phenomenon in electromagnetism, such as electromagnetic induction, light propagation, or the functioning of an electric circuit, happens when these quantities change.
• Physical law: describe the necessary and stable relationships between physical quantities.
  • Maxwell’s equation, relates $E(x,t)$ and $B(x,t)$
    • Ampere’s Law (with Maxwell’s correction): $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
    • Faraday’s Law of Induction: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
    • Gauss’s Law for Magnetism, Electricity:$\nabla \cdot \mathbf{B} = 0$, $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$
  • Lorentz Force Law: $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$
  • Ohm’s law: $V=IR$
  • Electromagnetic Wave equation: $\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$
  • Poynting Theorem: $\mathbf{S} = \mathbf{E} \times \mathbf{H}$
  • Wave Equation for Electromagnetic Waves: $\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$

How the State Changes (Physical Phenomena):

• A moving charge creates a magnetic field ($B$).
• A changing magnetic field induces an electric field (Faraday’s Law).
• A changing electric field can produce a magnetic field (Maxwell’s correction to Ampère’s Law).
• A charge in an electric or magnetic field experiences a force (Lorentz Force).
• A physical phenomenon in electromagnetism, such as electromagnetic induction, light propagation, or the functioning of an electric circuit, happens when these quantities change.

References

1. Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.
2. Jackson, J. D. (2021). Classical electrodynamics. John Wiley & Sons.
3. Greiner, W. (2012). Classical electrodynamics. Springer Science & Business Media.